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Exotic spheres and topological modular forms
Mark Behrens (MIT)
(joint with Mike Hill, Mike Hopkins, and Mark Mahowald)
Fantastic survey of the subject:
Milnor, “Differential topology: 46 years later”
(Notices of the AMS, June/July 2011)
http://www.ams.org/notices/201106/
Poincaré Conjecture
Q: Is every homotopy n-sphere homeomorphic
to an n-sphere?
A: Yes!
• 𝑛 = 2: easy.
• 𝑛 ≥ 5: (Smale, 1961) h-cobordism theorem
• 𝑛 = 4: (Freedman, 1982)
• 𝑛 = 3: (Perelman, 2003)
Smooth Poincaré Conjecture
Q: Is every homotopy n-sphere diffeomorphic to an n-sphere? A: Depends on n. • 𝑛 = 2: True - easy. • 𝑛 = 7: (Milnor, 1956) False – produced a smooth manifold
which was homeomorphic but not diffeomorphic to 𝑆7! [exotic sphere]
• 𝑛 ≥ 5: (Kervaire-Milnor, 1963) – `often’ false. (true for 𝑛 = 5,6). • 𝑛 = 3: (Perelman, 2003) True. • 𝑛 = 4: Unknown.
Smooth Poincaré Conjecture
Q: Is every homotopy n-sphere diffeomorphic to an n-sphere? A: Depends on n. • 𝑛 = 2: True - easy. • 𝑛 = 7: (Milnor, 1956) False – produced a smooth manifold
which was homeomorphic but not diffeomorphic to 𝑆7! [exotic sphere]
• 𝑛 ≥ 5: (Kervaire-Milnor, 1963) – `often’ false. (true for 𝑛 = 5,6). • 𝑛 = 3: (Perelman, 2003) True. • 𝑛 = 4: Unknown.
Main Question
For which n do there exist exotic n-spheres?
Kervaire-Milnor
Θ𝑛 ≔ {oriented smooth homotopy n-spheres}/h-cobordism
(note: if 𝑛 ≠ 4, h-cobordant ⇔ oriented diffeomorphic)
For 𝑛 ≢ 2(4):
0 → Θ𝑛𝑏𝑝
→ Θ𝑛 →𝜋𝑛
𝑠
𝐼𝑚 𝐽→ 0
Kervaire-Milnor
Θ𝑛 ≔ {smooth homotopy n-spheres}/h-cobordism
(note: if 𝑛 ≠ 4, h-cobordant ⇔ diffeomorphic)
For 𝑛 ≢ 2(4):
0 → Θ𝑛𝑏𝑝
→ Θ𝑛 →𝜋𝑛
𝑠
𝐼𝑚 𝐽→ 0
Kervaire-Milnor
Θ𝑛 ≔ {smooth homotopy n-spheres}/h-cobordism
For 𝑛 ≢ 2(4):
0 → Θ𝑛𝑏𝑝
→ Θ𝑛 →𝜋𝑛
𝑠
𝐼𝑚 𝐽→ 0
Θ𝑛𝑏𝑝
= subgroup of those which bound a
parallelizable manifold
𝜋𝑛𝑠 = stable homotopy groups of spheres
= 𝜋𝑛+𝑘 𝑆𝑘 for 𝑘 ≫ 0
𝐽: 𝜋𝑛 𝑆𝑂 → 𝜋𝑛𝑠 is the J-homomorphism.
Kervaire-Milnor
Θ𝑛 ≔ {smooth homotopy n-spheres}/h-cobordism
For 𝑛 ≢ 2(4):
0 → Θ𝑛𝑏𝑝
→ Θ𝑛 →𝜋𝑛
𝑠
𝐼𝑚 𝐽→ 0
Θ𝑛𝑏𝑝
= subgroup of those which bound a
parallelizable manifold
𝜋𝑛𝑠 = stable homotopy groups of spheres
= 𝜋𝑛+𝑘 𝑆𝑘 for 𝑘 ≫ 0
𝐽: 𝜋𝑛 𝑆𝑂 → 𝜋𝑛𝑠 is the J-homomorphism.
Kervaire-Milnor
Θ𝑛 ≔ {smooth homotopy n-spheres}/h-cobordism
For 𝑛 ≢ 2(4):
0 → Θ𝑛𝑏𝑝
→ Θ𝑛 →𝜋𝑛
𝑠
𝐼𝑚 𝐽→ 0
Θ𝑛𝑏𝑝
= subgroup of those which bound a
parallelizable manifold
𝜋𝑛𝑠 = stable homotopy groups of spheres
= 𝜋𝑛+𝑘 𝑆𝑘 for 𝑘 ≫ 0
𝐽: 𝜋𝑛 𝑆𝑂 → 𝜋𝑛𝑠 is the J-homomorphism.
Kervaire-Milnor
Θ𝑛 ≔ {smooth homotopy n-spheres}/h-cobordism
For 𝑛 ≢ 2(4):
0 → Θ𝑛𝑏𝑝
→ Θ𝑛 →𝜋𝑛
𝑠
𝐼𝑚 𝐽→ 0
For 𝑛 ≡ 2 4 :
0 → Θ𝑛𝑏𝑝
→ Θ𝑛 →𝜋𝑛
𝑠
𝐼𝑚 𝐽→ ℤ
2 → Θ𝑛−1𝑏𝑝
→ 0
Θ𝑛𝑏𝑝
• Trivial for n even
• Cyclic for n odd
Θ𝑛𝑏𝑝
• Trivial for n even
• Cyclic for n odd – Generated by boundary of an explicit parallelizable
manifold given by plumbing construction
Θ𝑛𝑏𝑝
• Trivial for n even
• Cyclic for n odd:
Θ𝑛𝑏𝑝
=
22𝑘 22𝑘+1 − 1 𝑛𝑢𝑚4𝐵𝑘+1
𝑘 + 1, 𝑛 = 4𝑘 + 3
ℤ2 , 𝑛 ≡ 1(4), ∃𝑀𝑛+1 𝑤𝑖𝑡ℎ Φ𝐾 = 1
0, 𝑛 ≡ 1 4 , ∄𝑀𝑛+1 𝑤𝑖𝑡ℎ Φ𝐾 = 1
Upshot: n even ⇒ bp gives no exotic spheres
𝑛 ≡ 3 (4) ⇒ bp gives exotic spheres (𝑛 ≥ 7)
𝑛 ≡ 1 (4) ⇒ bp gives exotic sphere only if there are no 𝑀𝑛+1 with Φ𝐾 = 1
J-homomorphism
𝐽: 𝜋𝑛𝑆𝑂 → 𝜋𝑛𝑠 ≅ Ω𝑛
𝑓𝑟
Given 𝛼: 𝑆𝑛 → 𝑆𝑂, apply it pointwise to the standard stable framing of 𝑆𝑛 to obtain a non-standard stable framing of 𝑆𝑛. Homotopy spheres are stably parallelizable, but not uniquely so – only get a well defined map
Θ𝑛 →𝜋𝑛
𝑠
𝐼𝑚 𝐽
J-homomorphism
𝐽: 𝜋𝑛𝑆𝑂 → 𝜋𝑛𝑠 ≅ Ω𝑛
𝑓𝑟
𝜋∗𝑠
Stable homotopy groups:
𝜋𝑛𝑠 ≔ lim
𝑘→∞𝜋𝑛+𝑘(𝑆
𝑘) (finite abelian groups for 𝑛 > 0)
Primary decomposition:
𝜋𝑛𝑠 = (𝜋𝑛
𝑠)(𝑝)𝑝 𝑝𝑟𝑖𝑚𝑒 e.g.: 𝜋3𝑠 = ℤ24 = ℤ8 ⊕ ℤ3
• Each dot represents a factor of 2, vertical lines indicate additive extensions
e.g.: (𝜋3𝑠)(2) = ℤ8, (𝜋8
𝑠)(2) = ℤ2 ℤ2
• Vertical arrangement of dots is arbitrary, but meant to suggest patterns
19
Computation: Mahowald-Tangora-Kochman Picture: A. Hatcher
• Each dot represents a factor of 2, vertical lines indicate additive extensions
e.g.: (𝜋3𝑠)(2) = ℤ8, (𝜋8
𝑠)(2) = ℤ2 ℤ2
• Vertical arrangement of dots is arbitrary, but meant to suggest patterns
20
21
Computation: Nakamura -Tangora Picture: A. Hatcher
(ns)(5)
n
22
Computation: D. Ravenel Picture: A. Hatcher
23
Adams spectral sequence
𝐸𝑥𝑡𝐴𝑠,𝑡(ℤ 𝑝 , ℤ 𝑝) ⇒ 𝜋𝑡−𝑠
𝑠𝑝
𝜋𝑡−𝑠𝑠
24
Adams spectral sequence
𝐸𝑥𝑡𝐴𝑠,𝑡(ℤ 𝑝 , ℤ 𝑝) ⇒ 𝜋𝑡−𝑠
𝑠𝑝
𝜋𝑡−𝑠𝑠
-Many differentials -𝑑𝑟 differentials go back by 1 and up by r
25
Adams spectral sequence
𝐸𝑥𝑡𝐴𝑠,𝑡(ℤ 𝑝 , ℤ 𝑝) ⇒ 𝜋𝑡−𝑠
𝑠𝑝
𝜋𝑡−𝑠𝑠
26
Adams spectral sequence
𝐸𝑥𝑡𝐴𝑠,𝑡(ℤ 𝑝 , ℤ 𝑝) ⇒ 𝜋𝑡−𝑠
𝑠𝑝
𝜋𝑡−𝑠𝑠
Kervaire Invariant
Φ𝐾: 𝜋𝑛𝑠 → ℤ 2
Browder:
(Φ𝐾≠ 0) ⇒ 𝑛 = 2𝑘 − 2
27
Kervaire Invariant
Φ𝐾: 𝜋𝑛𝑠 → ℤ 2
Browder:
(Φ𝐾 𝑥 ≠ 0) ⇔ 𝑥 𝑑𝑒𝑡𝑒𝑐𝑡𝑒𝑑 𝑏𝑦 ℎ𝑗2
𝑖𝑛 𝐴𝑆𝑆
28
Kervaire Invariant
Φ𝐾: 𝜋𝑛𝑠 → ℤ 2
Browder:
(Φ𝐾 𝑥 ≠ 0) ⇔ 𝑥 𝑑𝑒𝑡𝑒𝑐𝑡𝑒𝑑 𝑏𝑦 ℎ𝑗2
𝑖𝑛 𝐴𝑆𝑆
Computation in ASS: Φ𝐾 ≠ 0 for 𝑛 ∈ {2, 6, 14, 30, 62}
29
Kervaire Invariant
Φ𝐾: 𝜋𝑛𝑠 → ℤ 2
Browder:
(Φ𝐾 𝑥 ≠ 0) ⇔ 𝑥 𝑑𝑒𝑡𝑒𝑐𝑡𝑒𝑑 𝑏𝑦 ℎ𝑗2
𝑖𝑛 𝐴𝑆𝑆
Computation in ASS: Φ𝐾 ≠ 0 for 𝑛 ∈ {2, 6, 14, 30, 62}
Hill-Hopkins-Ravenel:
Φ𝐾 = 0 for all 𝑛 ≥ 254
(Note: the case of 𝑛 = 126 is still open) 30
Summary: Exotic spheres
Θ𝑛 ≠ 0 if:
• Θ𝑛𝑏𝑝
≠ 0:
o 𝑛 ≡ 3 (4) and 𝑛 ≥ 7
o 𝑛 ≡ 1 (4) and 𝑛 ∉ {1,5,13,29,61,125? } [Kervaire]
• Remains to check: is 𝜋𝑛
𝑠
𝐼𝑚 𝐽≠ 0 for
o 𝑛 even
o𝑛 ∈ {1,5,13,29,61,125? }
31
32
Summary: Exotic spheres
Θ𝑛 ≠ 0 if:
• Θ𝑛𝑏𝑝
≠ 0: o 𝑛 ≡ 3 (4) and 𝑛 ≥ 7 o 𝑛 ≡ 1 (4) and 𝑛 ∉ {1,5,13,29,61,125? }
•𝜋𝑛
𝑠
𝐼𝑚 𝐽≠ 0 for 𝑛 ≡ 2 (8)
• Remains to check: is 𝜋𝑛
𝑠
𝐼𝑚 𝐽≠ 0 for
o 𝑛 ≡ 0 (4) or 𝑛 ≡ −2 (8)
o𝑛 ∈ {1,5,13,29,61,125? }
33
Low dimensional computations
• Limitation: only know 𝜋𝑛𝑠
2 for 𝑛 ≤ 63
•𝜋𝑛
𝑠
𝐼𝑚 𝐽 𝑝= 0 in this range for 𝑝 ≥ 7.
34
Low dimensional computations
35
Stem p = 2 p = 3 p = 5
4 0 0 0
8 e 0 0
12 0 0 0
16 h4 0 0
20 κbar β1^2 0
24 h4 ε η 0 0
28 ε κbar 0 0
32 q 0 0
36 t β2 β1 0
40 κbar^2 β1^4 0
44 g2 0 0
48 e0 r 0 0
52 κbar q β2^2 0
56 κbar t 0 0
60 kbar^3 0 0
Non-trivial elements in 𝐶𝑜𝑘𝑒𝑟 𝐽: 𝑛 ≡ 0 (4)
Low dimensional computations
36
Stem p = 2 p = 3 p = 5
6 ν^2 0 0
14 k 0 0
22 ε k 0 0
30 θ4 β1^3 0
38 y β3/2 β1
46 w η β2 β1^2 0
54 v2^8 ν^2 0 0
62 h5 n β2^2 β1 0
Non-trivial elements in 𝐶𝑜𝑘𝑒𝑟 𝐽: 𝑛 ≡ −2 (8)
Low dimensional computations
37
Stem p = 2 p = 3 p = 5
1 0 0 0
5 0 0 0
13 0 β1 a1 0
29 0 β2 a1 0
61 0 β4 a1 0
Non-trivial elements in 𝐶𝑜𝑘𝑒𝑟 𝐽:
𝑛 ∈ {1,5,13,29,61} [where Θ𝑛𝑏𝑝
= 0 because of Kervaire classes]
Low dimensional computations
Conclusion
For 𝑛 ≤ 63, the only 𝑛 for which Θ𝑛 = 0 are: 1,2,3,4,5,6,12,61
38
Beyond low dimensions…
Strategy: try to demonstrate Coker J is non-zero in certain dimensions by producing infinite periodic families such as the one above.
Need to study periodicity in 𝜋∗𝑠
39
Periodicity in 𝜋∗𝑠
40
Periodicity in 𝜋∗𝑠
41
Periodicity in 𝜋∗𝑠
42
Periodicity in 𝜋∗𝑠
43
Periodicity in 𝜋∗𝑠
44
Periodicity in 𝜋∗𝑠
45
Periodicity in 𝜋∗𝑠
46
Periodicity in 𝜋∗𝑠
47
Periodicity in 𝜋∗𝑠
48
Periodicity in 𝜋∗𝑠
49
Periodicity in 𝜋∗𝑠
50
Periodicity in 𝜋∗𝑠
51
Periodicity in 𝜋∗𝑠
52
Periodicity in 𝜋∗𝑠
53
Periodicity in 𝜋∗𝑠
54
Periodicity in 𝜋∗𝑠
55
Periodicity in 𝜋∗𝑠
56
(ns)(5)
v1 - periodic layer consists solely of a-family
period = 2(p-1) = 8
57
(ns)(5)
v1 - periodic layer consists solely of a-family
period = 2(p-1) = 8
58
Greek letter notation: the 𝛼-family
59
(ns)(5)
v2 - periodic layer = b-family
period = 2(p2 - 1) = 48
60
(ns)(5)
61
𝑣1-torsion in the 𝑣2-family
62
Greek Letter Names (Miller-Ravenel-Wilson) 63
(ns)(5)
period = 2(p3 - 1) = 248
v3 - periodic layer = g-family
64
• 𝑣1-periodicity – completely understood
• 𝑣2-periodicity – know a lot for 𝑝 ≥ 5
– Knowledge for 𝑝 = 2,3 is subject of current research.
– For Θ𝑛, we will see 𝑝 = 2 dominates the discussion
• 𝑣3-periodicity – know next to nothing!
65
66
Exotic spheres from 𝛽-family
• 𝛽𝑘 = 𝛽𝑘/1,1 exists for 𝑝 ≥ 5 and 𝑘 ≥ 1
[Smith-Toda]
Θ𝑛 ≠ 0 for 𝑛 ≡ −2 𝑝 − 1 − 2 𝑚𝑜𝑑 2(𝑝2 − 1)
67
68
Coker J
n = 0 mod 4 n = -2 mod 8 (including Kervaire Inv 1) n = 2^k - 3 (where Θ_n^bp = 0 because of Kervaire class)
Stem p = 2 p = 3 p = 5 Stem p = 2 p = 3 p = 5 Stem p = 2 p = 3 p = 5
4 0 0 0 6 ν^2 0 0 1 0 0 0
8 e 0 0 14 k 0 0 5 0 0 0
12 0 0 0 22 ε k 0 0 13 0 β1 a1 0
16 h4 0 0 30 θ4 β1^3 0 29 0 β2 a1 0
20 κbar β1^2 0 38 y β3/2 β1 61 0 β4 a1 0
24 h4 ε η 0 0 46 w η β2 β1^2 0 125? 0
28 ε κbar 0 0 54 v2^8 ν^2 0 0
32 q 0 0 62 h5 n β2^2 β1 0
36 t β2 β1 0 70 0 0
40 κbar^2 β1^4 0 78 β2^3 0
44 g2 0 0 86 β6/2 β2
48 e0 r 0 0 94 β5 0
52 κbar q β2^2 0 102 β6/3 β1^2 0
56 κbar t 0 0 110 0
60 kbar^3 0 0 118 0
64 0 0 126 0
68 <a1,β3/2,β2> 0 134 β3
72 β2^2 β1^2 0 142 0
76 0 β1^2 150 0
80 0 0 158 0
84 β5 β1 0 166 0
88 0 0 174 0
92 β6/3 β1 0 182 β4
96 0 0 190 β1^5
100 β2 β5 0 198 0
104 0 206 β5/4
108 0 214 β5/3
112 0 222 β5/2
116 0 230 β5
120 0 238 β2 β1^4
124 β2 β1 246 0
128 0 254 0
132 0 262 0
136 0 270 0
140 0 278 β1
144 0 286 β3 β1^4
148 0 294 0
152 β1^4 302 0
156 0 310 0
160 0 318 0
Cohomology theories
• Use homology/cohomology to study homotopy
• A cohomology theory is a contravariant functor
𝐸: {Topological spaces} {graded ab groups}
𝑋 𝐸∗(𝑋)
• Homotopy invariant:𝑓 ≃ 𝑔 ⇒ 𝐸 𝑓 = 𝐸(𝑔)
• Excision: 𝑍 = 𝑋 ∪ 𝑌 (CW complexes)
⋯ → 𝐸∗ 𝑍 → 𝐸∗ 𝑋 ⊕ 𝐸∗ 𝑌 → 𝐸∗ 𝑋 ∩ 𝑌 →
69
Cohomology theories
• Use homology/cohomology to study homotopy
• A cohomology theory is a contravariant functor
𝐸: {Topological spaces} {graded ab groups}
𝑋 𝐸∗(𝑋)
• Homotopy groups:
𝜋𝑛 𝐸 ≔ 𝐸−𝑛(𝑝𝑡)
(Note, in the above, n may be negative)
70
Cohomology theories
• Example: singular cohomology – 𝐸𝑛 𝑋 = 𝐻𝑛(𝑋)
– 𝜋𝑛 𝐻 = ℤ, 𝑛 = 0,0, else.
• Example: Real K-theory – 𝐾𝑂0 𝑋 = 𝐾𝑂 𝑋 = Grothendieck group of ℝ-vector bundles over 𝑋.
– 𝜋∗𝐾𝑂 = (ℤ, ℤ 2 , ℤ 2 , 0, ℤ, 0, 0, 0, ℤ, ℤ 2 , ℤ 2 , 0, ℤ, 0, 0, 0… )
71
Hurewicz Homomorphism
• A cohomology theory E is a (commutative) ring theory if
its associated cohomology theory has “cup products”
𝐸∗(𝑋) is a graded commutative ring
• Such cohomology theories have a Hurewicz
homomorphism:
ℎ𝐸: 𝜋∗𝑠 → 𝜋∗𝐸
Example: 𝐻 detects 𝜋0𝑠 = ℤ.
72
Example: KO (real K-theory)
73
• To get more elements of Θ𝑛, need to start
looking at 𝑣2-periodic homotopy.
• Need a cohomology theory which sees a
bunch of 𝑣2-periodic classes in its
Hurewicz homomorphism
• 𝑡𝑚𝑓∗ 𝑋 - topological modular forms!
74
Topological Modular Forms
KO
• 𝑣1-periodic – 8-periodic
• Multiplicative group
• Bernoulli numbers
TMF
• 𝑣2-periodic – 576-periodic
• Elliptic curves
• Eisenstein series
(modular forms)
192-periodic at 𝑝 = 2
144-periodic at 𝑝 = 3
75
Topological Modular Forms
• There is a descent spectral sequence:
𝐻𝑠 ℳ𝑒𝑙𝑙; 𝜔⊗𝑡 ⇒ 𝜋2𝑡−𝑠𝑇𝑀𝐹
• Edge homomorphism:
𝜋2𝑘𝑇𝑀𝐹 → Ring of integral modular forms
(rationally this is an iso)
• 𝜋∗𝑇𝑀𝐹 has a bunch of 2 and 3-torsion, and the descent spectral sequence is highly non-trivial at these primes.
76
𝐻𝑠 ℳ𝑒𝑙𝑙; 𝜔⊗𝑡 ⇒ 𝜋2𝑡−𝑠𝑇𝑀𝐹 The decent spectral sequence for TMF
(p=2)
77
Exotic spheres from 𝛽-family
• 𝛽𝑘 = 𝛽𝑘/1,1 exists for 𝑝 ≥ 5 and 𝑘 ≥ 1
[Smith-Toda]
Θ𝑛 ≠ 0 for 𝑛 ≡ −2 𝑝 − 1 − 2 𝑚𝑜𝑑 2(𝑝2 − 1)
• 𝛽𝑘 exists for 𝑝 = 3 and 𝑘 ≡ 0,1,2,3,5,6 9
[B-Pemmaraju]
Θ𝑛 ≠ 0 for 𝑛 ≡ −6, 10, 26, 42, 74, 90 mod 144
78
Exotic spheres from 𝛽-family
• 𝛽𝑘 = 𝛽𝑘/1,1 exists for 𝑝 ≥ 5 and 𝑘 ≥ 1
[Smith-Toda]
Θ𝑛 ≠ 0 for 𝑛 ≡ −2 𝑝 − 1 − 2 𝑚𝑜𝑑 2(𝑝2 − 1)
• 𝛽𝑘 exists for 𝑝 = 3 and 𝑘 ≡ 0,1,2,3,5,6 9
[B-Pemmaraju]
Θ𝑛 ≠ 0 for 𝑛 ≡ −6, 10, 26, 42, 74, 90 mod 144
79
Hurewicz image of TMF (p = 3)
80
81
Coker J
n = 0 mod 4 n = -2 mod 8 (including Kervaire Inv 1) n = 2^k - 3 (where Θ_n^bp = 0 because of Kervaire class)
Stem p = 2 p = 3 p = 5 Stem p = 2 p = 3 p = 5 Stem p = 2 p = 3 p = 5
4 0 0 0 6 ν^2 0 0 1 0 0 0
8 e 0 0 14 k 0 0 5 0 0 0
12 0 0 0 22 ε k 0 0 13 0 β1 a1 0
16 h4 0 0 30 θ4 β1^3 0 29 0 β2 a1 0
20 κbar β1^2 0 38 y β3/2 β1 61 0 β4 a1 0
24 h4 ε η 0 0 46 w η β2 β1^2 0 125? 0
28 ε κbar 0 0 54 v2^8 ν^2 0 0
32 q 0 0 62 h5 n β2^2 β1 0
36 t β2 β1 0 70 0 0
40 κbar^2 β1^4 0 78 β2^3 0
44 g2 0 0 86 β6/2 β2
48 e0 r 0 0 94 β5 0
52 κbar q β2^2 0 102 β6/3 β1^2 0
56 κbar t 0 0 110 0
60 kbar^3 0 0 118 0
64 0 0 126 0
68 <a1,β3/2,β2> 0 134 β3
72 β2^2 β1^2 0 142 0
76 0 β1^2 150 0
80 0 0 158 0
84 β5 β1 0 166 0
88 0 0 174 β1^3 0
92 β6/3 β1 0 182 β3/2 β4
96 0 0 190 β2 β1^2 β1^5
100 β2 β5 0 198 0
104 0 206 β2^2 β1 β5/4
108 0 214 β5/3
112 0 222 β2^3 β5/2
116 0 230 β6/2 β5
120 0 238 β5 β2 β1^4
124 β2 β1 246 β6/3 β1^2 0
128 0 254 0
132 0 262 0
136 0 270 0
140 0 278 β1
144 0 286 β3 β1^4
148 0 294 0
152 β1^4 302 0
156 0 310 0
160 0 318 β1^3 0
𝑣2-periodicity at the prime 2
82
𝑣2-periodicity at the prime 2
83
Thm: (B-Mahowald)
The complete Hurewicz image:
The decent spectral sequence for TMF
(p=2) 84
Hurewicz image of TMF (p = 2)
85
86
Coker J
n = 0 mod 4 n = -2 mod 8 (including Kervaire Inv 1) n = 2^k - 3 (where Θ_n^bp = 0 because of Kervaire class)
Stem p = 2 p = 3 p = 5 Stem p = 2 p = 3 p = 5 Stem p = 2 p = 3 p = 5
4 0 0 0 6 ν^2 0 0 1 0 0 0
8 e 0 0 14 k 0 0 5 0 0 0
12 0 0 0 22 ε k 0 0 13 0 β1 a1 0
16 h4 0 0 30 θ4 β1^3 0 29 0 β2 a1 0
20 κbar β1^2 0 38 y β3/2 β1 61 0 β4 a1 0
24 h4 ε η 0 0 46 w η β2 β1^2 0 125? w kbar^4 0
28 ε κbar 0 0 54 v2^8 ν^2 0 0 = in tmf
32 q 0 0 62 h5 n β2^2 β1 0 = not in tmf, not known to be v2-periodic
36 t β2 β1 0 70 <kbar w,ν,η> 0 0 = not in tmf, but v2-periodic
40 κbar^2 β1^4 0 78 β2^3 0 = Kervaire
44 g2 0 0 86 β6/2 β2 = trivial
48 e0 r 0 0 94 β5 0
52 κbar q β2^2 0 102 v2^16 ν^2 β6/3 β1^2 0
56 κbar t 0 0 110 v2^16 k 0
60 kbar^3 0 0 118 v2^16 η^2 kbar 0
64 0 0 126 0
68 v2^8 k ν^2 <a1,β3/2,β2> 0 134 β3
72 β2^2 β1^2 0 142 v2^16 η w 0
76 0 β1^2 150 (v2^16 ε kbar)η^2 v2^9 0
80 kbar^4 0 0 158 0
84 β5 β1 0 166 0
88 0 0 174 beta32/8 β1^3 0
92 β6/3 β1 0 182 beta32/4 β3/2 β4
96 0 0 190 β2 β1^2 β1^5
100 kbar^5 β2 β5 0 198 v2^32 ν^2 0
104 v2^16 ε 0 206 k β2^2 β1 β5/4
108 0 214 ε k β5/3
112 0 222 β2^3 β5/2
116 2v2^16 kbar 0 230 β6/2 β5
120 0 238 w η β5 β2 β1^4
124 v2^16 k^2 β2 β1 246 v2^8 ν^2 β6/3 β1^2 0
128 v2^16 q 0 254 0
132 0 262 <kbar w,ν,η> 0
136 <v2^16 k kbar,2,ν^2> 0 270 0
140 0 278 β1
144 v2^9 0 286 β3 β1^4
148 v2^16 ε kbar 0 294 v2^16 ν^2 v2^18 0
152 β1^4 302 v2^16 k 0
156 <Δ^6 ν^2,2ν,η^2> 0 310 v2^16 η^2 kbar 0
160 0 318 β1^3 0